Re: Suggestion: Visualization of complex functions with Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg44862] Re: Suggestion: Visualization of complex functions with Mathematica
- From: "Steve Luttrell" <luttrell at _removemefirst_westmal.demon.co.uk>
- Date: Thu, 4 Dec 2003 03:04:19 -0500 (EST)
- References: <bqkb3v$hkt$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Have a look at the Graphics`ComplexMap` package. It is documented in the Add-ons/StandardPackages/Graphics/ComplexMap entry in the Help browser. -- Steve Luttrell West Malvern, UK ""René Böhlendorf"" <R.Boehlendorf at t-online.de> wrote in message news:bqkb3v$hkt$1 at smc.vnet.net... > Dear Madams and Sirs, > > I am an owner of Mathematica 4.2.0.0 and got > my diploma in mathematics from Technische Universit=E4t Berlin in the > year 1981. My thesis was about complex analysis with the title 'Der > universelle Teichm=FCllerraum', which is a space of conformal mappings > from a part of the complex sphere with quasiconformal continuations to > the complete complex sphere. > > I was always interested to visualize complex functions, but the possi- > bilities are rather limited as we can visually imagine only threedimen- > sional objects and the graph of a complex function of one variable > would be a fourdimensional object. > > I was wondering if your program Mathematica would try to offer some- > thing to visualize complex functions in terms of four dimensions, but > I did not find something which is quite understandable. > > Nevertheless I want to offer two ideas to you which you could evaluate > and possibly implement in future versions of Mathematica. I understand > that Mathematica aims mainly at teaching mathematics besides being a > general help for technicians, physicists, mathematicians and all other > persons applying mathematics. > > Visualizations of complex functions are up to now only graphs of the > absolute value, the real or the imaginary part of the function. This > gives only a poor impression similar to seeing only a shadow of an > object instead of the object itself. > > 1. My first idea is to view the complex plane as a normal twodimensio- > nal Euclidian plane and attach a twodimensional vector to each point > to represent the value of a complex function there. > > Of course there are many problems to view such an object. It would > look more or less like a cornfield and it would be difficult to get > a good impression of the behaviour of the complex function which is > displayed. > > I thought about drawing such a picture with help of Mathematica or > other programs for some simple functions and use only a small set of > points, but found it too troublesome for me and of no general value. > > A function in Mathematica which would draw such a 'cornfield' auto- > matically using a raster would save me this trouble and be a great > fun for me! Although it would be not too interesting for professio- > nal mathematicians who work in complex analysis it could serve as a > good help to teach complex analysis for beginners by showing true > graphs of a complex function. > > 2. My second idea is to use colour and it is perhaps the better one. > > The complex plane or sphere would be coloured in such a way that > different regions have different colours, perhaps even using con- > tinuously changing colours. > > The visualization of the complex function would be a second plane > or sphere showing the same colours as the first one, but moved to > the place they get by applying the complex function. I believe that > this will produce good and impressive pictures for many complex > functions although I did never see such pictures up to now! > > As I worked in the area of software development since 1981 I am able > to estimate that quite a lot of work would have to be done if you would > decide to implement one or both of my ideas in Mathematica. > > At least I hope that you will think the ideas over even if you should > decide not to implement them. A few lines as a response to me when it > is convenient for you would make me happy! > > Faithfully yours, > Ren=E9 B=F6hlendorf >